The deductive closure of a set of sentences is trivial, i.e., it includes every well-formed sentence, if this set contains a contradiction and the consequence relation employed is either classical or intuitionistic. Over the past few decades, a number of paraconsistent logics, or logics specifically designed not to trivialize inconsistent theories, have been developed. The present work investigates philosophical issues arising from the development of paraconsistent formal systems. In the introductory chapter, as well as on a chapter that extracts learnings from Wittgenstein's career-long preoccupation with contradiction, I endeavor to determine just what is the problem with contradictions, as they arise in both natural and formal languages. I then consider in detail two kinds of paraconsistent logic: their formal characteristics, the motivation for their formulation, their possible applications, and objections that may be raised against them. Special attention is devoted to a logical system that deliberately permits the evaluation of certain contradictions as being true, as well as to the attendant philosophical position, known as dialetheism, according to which there are, in fact, true contradictions. I raise a number of objections to this strong (and resilient) form of paraconsisteney, which, taken together, constitute a rebuttal of the view, thus carrying out a task that a number of authors have signaled as pressing, but which has not so far been undertaken in detail in the literature.
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